Sure, let's simplify the given expression step by step.
The expression we need to simplify is:
[tex]\[
\frac{\sqrt{240 x^{11} y^9}}{\sqrt{5 x}}
\][/tex]
### Step 1: Combine the Radicals
First, let's combine the radicals in the numerator and the denominator.
[tex]\[
\frac{\sqrt{240 x^{11} y^9}}{\sqrt{5 x}} = \sqrt{\frac{240 x^{11} y^9}{5 x}}
\][/tex]
### Step 2: Simplify Inside the Radical
Now, let's simplify the expression inside the square root. Divide 240 by 5:
[tex]\[
\frac{240}{5} = 48
\][/tex]
Then, we simplify [tex]\(x^{11}\)[/tex] and [tex]\(x\)[/tex]:
[tex]\[
\frac{x^{11}}{x} = x^{10}
\][/tex]
Since [tex]\(y^9\)[/tex] does not have a counterpart in the denominator, it remains as is.
Putting it all together:
[tex]\[
\sqrt{\frac{240 x^{11} y^9}{5 x}} = \sqrt{48 x^{10} y^9}
\][/tex]
### Step 3: Break Down the Square Roots
Next, let's break down the square root of each factor:
[tex]\[
\sqrt{48 x^{10} y^9} = \sqrt{48} \cdot \sqrt{x^{10}} \cdot \sqrt{y^9}
\][/tex]
### Step 4: Simplify Each Radical
Simplify each part individually:
- [tex]\(\sqrt{48}\)[/tex]: We know that [tex]\(48 = 16 \times 3\)[/tex], and since [tex]\(\sqrt{16} = 4\)[/tex], we get:
[tex]\[
\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3}
\][/tex]
- [tex]\(\sqrt{x^{10}}\)[/tex]: For [tex]\(x^{10}\)[/tex], since [tex]\(10\)[/tex] is even:
[tex]\[
\sqrt{x^{10}} = x^{10/2} = x^5
\][/tex]
- [tex]\(\sqrt{y^9}\)[/tex]: For [tex]\(y^9\)[/tex], we can rewrite it as [tex]\(y^8 \cdot y\)[/tex] since [tex]\(8\)[/tex] is the largest even number less than or equal to [tex]\(9\)[/tex]:
[tex]\[
\sqrt{y^9} = \sqrt{y^8 \cdot y} = \sqrt{y^8} \cdot \sqrt{y} = y^{8/2} \cdot \sqrt{y} = y^4 \sqrt{y}
\][/tex]
### Step 5: Combine All Simplified Parts
Now, combine all simplified parts:
[tex]\[
\sqrt{48} \cdot \sqrt{x^{10}} \cdot \sqrt{y^9} = 4 \sqrt{3} \cdot x^5 \cdot y^4 \sqrt{y}
\][/tex]
Finally, combining these together:
[tex]\[
4 \sqrt{3} \cdot x^5 \cdot y^4 \cdot \sqrt{y} = 4 \sqrt{3} x^5 y^4 \sqrt{y}
\][/tex]
So, the completely simplified form of the given expression is:
[tex]\[
\frac{\sqrt{240 x^{11} y^9}}{\sqrt{5 x}} = 4 \sqrt{3} \frac{\sqrt{x^{11} y^9}}{\sqrt{x}} = 4 \sqrt{3}\frac{x^{5.5} y^{4.5}}{x^{0.5}} = 4 \sqrt{3} x^{5} y^{4} \sqrt{y}
\][/tex]
Hence our final result, simplified, is:
[tex]\[
\boxed{4 \sqrt{3} x^{5} y^{4} \sqrt{y}}
\][/tex]
